Non-Associative Flux Algebra in String and M-theory from ... Non-Associative Flux Algebra in String

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    Supergravity @ 40, GGI Florence, October 27th, 2016

    Non-Associative Flux Algebra in String and M-theory from Octonions

    DIETER LÜST (LMU, MPI)

    Donnerstag, 27. Oktober 16

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    Supergravity @ 40, GGI Florence, October 27th, 2016

    In collaboration with M. Günaydin & E. Malek, arXiv:1607.06474

    Non-Associative Flux Algebra in String and M-theory from Octonions

    DIETER LÜST (LMU, MPI)

    Donnerstag, 27. Oktober 16

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    This talk is dedicated to my friend Ioannis Bakas

    Donnerstag, 27. Oktober 16

  • Outline:

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    II) Non-associative R-flux algebra for closed strings

    I) Introduction

    III) R-flux algebra from octonions

    IV) M-theory up-lift of R-flux background

    V) Non-associative R-flux algebra in M-theory

    Donnerstag, 27. Oktober 16

  • Point particles in classical Einstein gravity „see“ continuous Riemannian manifolds.

    Geometry in general depends on, with what kind of objects you test it.

    Strings may see space-time in a different way.

    4

    We expect the emergence of a new kind of stringy geometry.

    I) Introduction

    - [xi, xj ] = 0

    Donnerstag, 27. Oktober 16

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    Closed strings in non-geometric R-flux backgrounds ⇒ non-associative phase space algebra:

    R. Blumenhagen, E. Plauschinn, arXiv:1010.1263.

    D.L., arXiv:1010.1361;⇥ x

    i , x

    j ⇤

    = i l

    3 s

    ~ R ijk

    pk

    ⇥ x

    i , p

    j ⇤

    = i~�ij , ⇥ p

    i , p

    j ⇤

    = 0

    =) ⇥ x

    i , x

    j , x

    k ⇤ ⌘ 1

    3 ⇥⇥

    x

    1 , x

    2 ⇤ , x

    3 ⇤ + cycl. perm. = l3sR

    ijk

    Donnerstag, 27. Oktober 16

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    Closed strings in non-geometric R-flux backgrounds ⇒ non-associative phase space algebra:

    R. Blumenhagen, E. Plauschinn, arXiv:1010.1263.

    D.L., arXiv:1010.1361;⇥ x

    i , x

    j ⇤

    = i l

    3 s

    ~ R ijk

    pk

    ⇥ x

    i , p

    j ⇤

    = i~�ij , ⇥ p

    i , p

    j ⇤

    = 0

    =) ⇥ x

    i , x

    j , x

    k ⇤ ⌘ 1

    3 ⇥⇥

    x

    1 , x

    2 ⇤ , x

    3 ⇤ + cycl. perm. = l3sR

    ijk

    This algebra can be derived from closed string CFT. R. Blumenhagen, A. Deser, D.L. , E. Plauschinn, F. Rennecke, arXiv:1106.0316

    C. Condeescu, I. Florakis, D. L., arXiv:1202.6366

    C. Blair, arXiv:1405.2283

    D. Andriot, M. Larfors, D.L. , P. Patalong:arXiv:1211.6437

    I. Bakas, D.L., arXiv:1505.04004

    Donnerstag, 27. Oktober 16

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    Closed strings in non-geometric R-flux backgrounds ⇒ non-associative phase space algebra:

    R. Blumenhagen, E. Plauschinn, arXiv:1010.1263.

    D.L., arXiv:1010.1361;⇥ x

    i , x

    j ⇤

    = i l

    3 s

    ~ R ijk

    pk

    ⇥ x

    i , p

    j ⇤

    = i~�ij , ⇥ p

    i , p

    j ⇤

    = 0

    =) ⇥ x

    i , x

    j , x

    k ⇤ ⌘ 1

    3 ⇥⇥

    x

    1 , x

    2 ⇤ , x

    3 ⇤ + cycl. perm. = l3sR

    ijk

    This algebra can be derived from closed string CFT. R. Blumenhagen, A. Deser, D.L. , E. Plauschinn, F. Rennecke, arXiv:1106.0316

    C. Condeescu, I. Florakis, D. L., arXiv:1202.6366

    C. Blair, arXiv:1405.2283

    D. Andriot, M. Larfors, D.L. , P. Patalong:arXiv:1211.6437

    I. Bakas, D.L., arXiv:1505.04004

    This algebra is also closely related to double field theory. R. Blumenhagen, M. Fuchs, F. Hassler, D.L. , R. Sun, arXiv:1312.0719

    Donnerstag, 27. Oktober 16

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    Two questions:

    Donnerstag, 27. Oktober 16

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    Two questions:

    On the mathematical side:

    How is the R-flux algebra related to other known non-associative algebras, in particular to the algebra of the octonions?

    Donnerstag, 27. Oktober 16

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    Two questions:

    On the mathematical side:

    How is the R-flux algebra related to other known non-associative algebras, in particular to the algebra of the octonions?

    Can one lift the R-flux algebra of closed strings to M-theory?

    On the physics side:

    Donnerstag, 27. Oktober 16

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    Two questions:

    On the mathematical side:

    How is the R-flux algebra related to other known non-associative algebras, in particular to the algebra of the octonions?

    Can one lift the R-flux algebra of closed strings to M-theory?

    On the physics side:

    Our conjecture: the answers to these two questions are closely related

    Donnerstag, 27. Oktober 16

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    II) Non-geometric string flux backgrounds

    Three-dimensional string flux backgrounds:

    Hijk Ti�! f ijk

    Tj�! Qijk Tk�! Rijk , (i, j, k = 1, . . . , 3)

    Chain of three T-duality transformations:

    ds

    2 = (dx1)2 + (dx2)2 + (dx3)2 , B12 = Nx3

    H123 = NH-flux:

    (Hellerman, McGreevy, Williams (2002); C. Hull (2004); Shelton, Taylor, Wecht (2005); Dabholkar, Hull, 2005)

    (i) with H-flux:T 3

    Donnerstag, 27. Oktober 16

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    ds

    2 = � dx

    1 �Nx3dx2 �2 + (dx2)2 + (dx3)2 , B2 = 0

    Globally defined 1-forms:

    1 = dx1 �Nx3dx2 , ⌘2 = dx2 , ⌘3 = dx3

    d⌘i = f ijk⌘ j ^ ⌘k

    Geometric flux: f123 = N

    x

    1(ii) Twisted torus tilde : T-duality along T̃ 3

    is a U(1) bundle over :T̃ 3 T 2

    Donnerstag, 27. Oktober 16

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    (iii) Q-flux background:

    ds

    2 = (dx1)2 + (dx2)2

    1 + N2 (x3)2 + (dx3)2 , B23 =

    Nx

    3

    1 + N2 (x3)2

    This background is globally not well defined, but it is patched together by a T-duality transformation.

    C. Hull (2004)

    T-duality along x

    2

    ⇒ T - fold

    Donnerstag, 27. Oktober 16

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    (iii) Q-flux background:

    ds

    2 = (dx1)2 + (dx2)2

    1 + N2 (x3)2 + (dx3)2 , B23 =

    Nx

    3

    1 + N2 (x3)2

    This background is globally not well defined, but it is patched together by T-duality transformation.

    To make it well defined use double field theory:

    Coordinates: (x1, x2, x3; x̃1, x̃2, x̃3)

    C. Hull (2004)

    W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...)

    SO(3,3) double field theory:

    T-duality along x

    2

    ⇒ T - fold

    Donnerstag, 27. Oktober 16

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    M. Grana, R. Minasian, M. Petrini, D. Waldram (2008); D. Andriot, O. Hohm, M. Larfors, D.L., P. Patalong (2011,2012); R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, C. Schmid (2013); D. Andriot, A. Betz (2013)

    The dual background can then by described by „dual“ metric and a bi-vector:

    Bij(x) ! �ij(x) = 1 2

    ⇣ (g �B)�1 � (g + B)�1

    ⌘ ,

    g(x) ! ĝ(x) = 1 2

    ⇣ (g �B)�1 + (g + B)�1

    ⌘�1 .

    T ij

    T ij

    Qijk = @k� ij

    Donnerstag, 27. Oktober 16

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    M. Grana, R. Minasian, M. Petrini, D. Waldram (2008); D. Andriot, O. Hohm, M. Larfors, D.L., P. Patalong (2011,2012); R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, C. Schmid (2013); D. Andriot, A. Betz (2013)

    The dual background can then by described by „dual“ metric and a bi-vector:

    d̂s

    2 = (dx1)2 + (dx2)2 + (dx3)2 , �12 = Nx3

    Q-flux: Q123 = N

    For the Q-flux background one obtains:

    Bij(x) ! �ij(x) = 1 2

    ⇣ (g �B)�1 � (g + B)�1

    ⌘ ,

    g(x) ! ĝ(x) = 1 2

    ⇣ (g �B)�1 + (g + B)�1

    ⌘�1 .

    T ij

    T ij

    Qijk = @k� ij

    Donnerstag, 27. Oktober 16

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    Then one obtains from the CFT of the Q-flux background the following commutation relation among the coordinates:

    ⇥ x

    1 , x

    2 ⇤

    = Np̃3

    In general:

    ⇥ x

    i , x

    j ⇤

    = i l

    2 s

    ~

    I

    S1k

    Q

    ij k (x) dx

    k = i l

    3 s

    ~ Q ij k p̃

    k

    winding number = dual momentum

    Sigma-model for non-geometric backgrounds: A. Chatzistavrakidis, L. Jonke, O. Lechtenfeld, arXiv:1505.05457

    I. Bakas, D.L., arXiv:1505.04004

    Donnerstag, 27. Oktober 16

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    (iv) R-flux background:

    Buscher rule fails and one would get a background that is even locally not well defined.

    T-duality along x3

    Donnerstag, 27. Oktober 16

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    (iv) R-flux background:

    Buscher rule fails and one would get a background that is even locally not well defined.

    T-duality along x3

    R-flux can be defined in double field theory:

    T k �

    ij(xk) ! �ij(x̃k)

    T k x

    k ! x̃k

    Rijk = 3@̂[k�ij]

    Donnerstag, 27. Oktober 16

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    Strong constraint of DFT is violated by this background.

    d̂s

    2 = (dx1)